Canberra: Malcolm Turnbull has declared he can lead the Coalition to the next federal election because he is capable of sorting out contentious issues such as same-sex marriage and energy, as parliament braces for the potential resignation of Jacqui Lambie over dual citizenship concerns.

Turnbull made a public appeal for calm when he faced travelling reporters in the Philippines on Monday, after being asked whether he was concerned this trip could be his last international meeting, and what message he would send to doubters in Coalition ranks.

Back in Canberra, the Senate was awash with speculation after a chaotic opening day on Monday, that Lambie was on the brink of resigning because her father was born in Scotland.

If the Tasmanian senator quits, as many Senate colleagues anticipate, she will join two Greens; the deputy leader of the National party, Fiona Nash; One Nation’s Malcolm Roberts; and the Senate president, the Liberal Stephen Parry, among senators who have fallen foul of the constitutional requirements under section 44.

Pressure on Turnbull has intensified after the latest Newspoll charted a five-point drop in the better prime minister rating, a metric where he normally dominates his opponent, Bill Shorten.

The new poll put the Liberal deputy leader, Julie Bishop, a potential rival in any leadership conflagration, ahead of Turnbull as preferred party leader, 40 per cent to 27 per cent, with Peter Dutton behind both on 11 per cent.

The poor result in the poll published late on Sunday night followed weeks of rolling chaos and contention courtesy of the citizenship imbroglio, culminating in the loss of the government’s lower-house majority at the weekend with the resignation of John Alexander.

Turnbull has also been on the receiving end of public criticism from internal enemies as the citizenship crisis has intensified, and he has recently suffered cabinet leaks about the government’s internal deliberations over a new disclosure regime which is likely to trigger a number of fresh referrals to the high court.

The prime minister said his message to people in the Coalition who might be doubting his capacity to get the government out of its political hole was: “We face plenty of challenges in politics and we deal with them.”

Turnbull said democracy was certainly unruly but that had always been the case. He said the record of his prime ministership was “when these big issues come up, resolving them, getting them sorted out”.

The prime minister said he was focused on delivering for Australians, on national security and economic security, and when it came to resolving the citizenship fracas, MPs and senators needed to comply with the law.

He said the disclosure regime would put all the facts on the table, and then the houses of parliament could move forward with any referrals to the high court that might be necessary.

With the Senate in session for a week, the government on Monday struck a deal with Labor over the disclosure regime for dual citizenships, with declarations from MPs to be in the public domain no later than 1 December.

It also achieved a largely smooth transition to a new Senate president. Scott Ryan, a Turnbull ally and former frontbencher, assumed the post on Monday with Labor’s support but found himself immediately tested during a rowdy session of question time.

Another internal pressure point also looms, with a renewed internal fight under way on marriage equality in anticipation of a yes vote in the postal survey. The result will be announced on Wednesday.